Prof. Dr. Charles Parsons

Some consequences of the entanglement of logic and mathematics

First-order logic has no substantial ontological commitments in that what is valid (or derivable) is true in any non-empty domain. But this is misleading in a way, because domains of cardinality up to the smallest infinite cardinality are needed to witness nonvalidity or failures of consequence. Domains of natural numbers are sufficient to witness nonvalidity, and mathematics seems necessary in the infinite cases. This entanglement of first-order logic and arithmetic seems to be resisted by John Etchemendy in some arguments in The Concept of Logical Consequence. Second-order logic is entangled with more powerful and problematic mathematics: set theory, including higher set theory. However, the issues are different depending on whether one attends primarily to proof procedures or to the "standard" semantics. It will be argued that some of the popularity of second-order logic in contemporary philosophy of mathematics reflects inattention or resistance to the implications of this entanglement. These considerations will be applied to the contested question of the status of plural logic.